In [1]}, Bright et al. provide a generalization of the original local to global principle to number fields. We will state the result translated to our setting. Let us define the \(\mathbb {R}\) -algebra \(k_\infty =\mathfrak {o}\otimes _\mathbb {Z} \mathbb {R}\) , which has covolume \({\Delta _K}^{1/2}\) , where \(\Delta _K\)  denotes the discriminant of \(K\)  and \(k_\infty \)  is endowed with the Haar measure \(\nu _{\infty }\) , see [2]}.
That is, for some set \(X \subseteq k_\infty \)  we have
\(\nu _\infty (X)= \frac{1}{{ \Delta _K}^{1/2}} \mu _\infty (f(X)),\) 
